Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{p^2 - 2p}{p^2 - 6p + 8}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{p^2 - 2p}{p^2 - 6p + 8} = \dfrac{(p)(p - 2)}{(p - 4)(p - 2)} $ Notice that the term $(p - 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p - 2)$ gives: $r = \dfrac{p}{p - 4}$ Since we divided by $(p - 2)$, $p \neq 2$. $r = \dfrac{p}{p - 4}; \space p \neq 2$